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A005934 Highly powerful numbers.
(Formerly M3333)
15
1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..300

R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

C. B. Lacampagne and J. L. Selfridge, Large highly powerful numbers are cubeful, Proc. Amer. Math. Soc., 91 (1984), 173-181.

Index entries for sequences related to powerful numbers

FORMULA

For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.

MATHEMATICA

a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *)

With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *)

PROG

(PARI) {prdex(n)=local(s, fac); s=1; fac=factor(n); for(k=1, matsize(fac)[1], s=s*fac[k, 2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1, m, if(prdex(n)>rec, rec=prdex(n); print1(n", ")))}

CROSSREFS

Cf. A036965, A001694, A007532, A005934, A005188, A003321, A014576, A023052, A046074.

Cf. A085629.

Sequence in context: A196875 A111073 A298807 * A085629 A233442 A046055

Adjacent sequences:  A005931 A005932 A005933 * A005935 A005936 A005937

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Hardy and Subbarao give an extensive table.

Corrected and extended by Jason Earls, Jul 10 2003

STATUS

approved

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Last modified January 21 17:47 EST 2019. Contains 319349 sequences. (Running on oeis4.)